Scientific program

Video of the talks, lectures and lecture courses


1. Schedule

2. Special events

  • September 17, 2013 at 17:00, Marble Hall, PDMI, nab. Fontanka 27

Joint meeting of St. Petersburg Mathematical Society and Cohomology in Mathematics and Physics program at Euler Institute

John W. Morgan

(Simons Center for Geometry and Physics, Stony Brook University, NY)

Low dimensional topology: a survey(video)

  • September 23, 2013 at 11:00, Euler Institute, Pesochnaya nab. 10

Joint meeting of St. Petersburg Mathematical Society and Cohomology in Mathematics and Physics program at Euler Institute

Dennis Sullivan

(CUNY Graduate Center & Stony Brook University, NY)

Finite algebraic models of processes in space(video)

  • September 23, 2013, 15:00-15:45, Euler Institute, Pesochnaya nab. 10

Nikolai Andreev

(Steklov Mathematical Institute, Moscow)

Mathematical études (3D animated films)- have fun!

  • September 23, 2013 at 17:00, Marble Hall, PDMI, nab. Fontanka 27

Joint meeting of St. Petersburg Mathematical Society and Cohomology in Mathematics and Physics program at Euler Institute

James H. Simons

(Stony Brook University, NY)

Roots of differential cohomology (video)

  • September 24, 2013 at 11:00, Euler Institute, Pesochnaya nab. 10

Ludwig D. Faddeev

Anomalies and cohomologies (video)

Anomaly is an object in the quantum theory of fields, physicists like fancy words. Field Theory is a system with infinite number of degrees of freedom and upon quantisation some symmetries break down. This is called anomaly.

I will argue that in the case of the Yang-Mills theory the symmetry is not broken but is rather modified. Group action is supplied with a factor and/or is substituted by a projective presentation. Corresponding one and two cocycles are produced by the Chern-Simons descent procedure starting from the third Chern class.

  • September 24, 2013 at 12:00, Euler Institute, Pesochnaya nab. 10

James Simons & Dennis Sullivan

Differential K-theory and its Characters (video)

One begins with how complex bundles with connections define, via the Chern-Simons equivalence relation on connections, the extension (called differential K-theory) of usual K-theory by all total odd forms modulo pull backs from the unitary group of the canonical primitive closed forms...

One also defines the analog of differential characters for these objects. A K-character assigns to each odd dimensional closed manifold with additional geometrical structure mapping into the base a number in R/Z. These numbers satisfy a deformation property and a product property.

The first theorem gives a bijection between K-characters and the elements of differential K-theory. One corollary is a differential geometric construction of the known (but unpublished) set of complete numerical invariants for elements in usual complex K-theory.

The first theorem also yields a natural push forward in differential K-theory for families of almost complex manifolds geometrized as a riemannian submersion. A non trivial computation shows that the natural product connection on the total space used to define the push forward is chern-simons equivalent to the limit of the levi civita connections on the total space as the fibre are scaled down to points.

Finally the stage is set to apply the Atiyah-Patodi-Singer theorem to obtain an analytic computation of the differential K-theory push forward at each value of the K-character as a limit of eta-invariants.

  • September 30, 2013, at Polit.ru

Leon Takhtajan

  • Polit.ru public lecture, October 3, 2013 at 19:00, cafe ZaVtra, Moscow

Leon Takhtajan

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3. Main lecture courses

(four-six lectures, 90 min = 2 x 45 min each)

  • Course 1. Cohomology: differential, etale and motivic cohomology theories I. Panin (video)
  • Course 2 . Chern-Simons theory, knot homology and quantum curves S. Gukov (video)
  • Course 3. Introduction to topological quantum field theories: Seiberg-Witten and Donaldson-Witten theories M. Marino & A. Losev (video)
  • Course 4 Mirror symmetry: quantum cohomology, Landau-Ginzburg models & D-branes D. Orlov (video)

4. Topics courses

(two-four lectures, 90 min = 2 x 45 min each)

  • Topics 1 Quantum invariants R. Kashaev (video)
  • Topics 2 Introduction to hyperkahler manifolds M. Verbitsky (video)
  • Topics 3 Geometric representation theory H. Nakajima (video)
  • Topics 4 Associators, Grothendieck-Teichmüller group and flat connections A. Alekseev (video)
  • Topics 5 Algebraic integrable systems and quantum Riemann surfaces I. Krichever & F. Smirnov (video)
  • Topics 6 Supersymmetric gauge theories and quantum integrability V. Pestun (video)
  • Topics 7 Quasi-maps, Uhlenbeck spaces and quantum K-theory A. Braverman (video)

5. Lectures

  • "Introduction to Riemannian geometry, curvature and Ricci flow, with applications to the topology of 3-dimensional manifolds" J. Morgan (video)
  • "Kahler-Einstein metrics and Ricci flow" G. Tian (video)

6. Lecturers include:

S. Gukov, R. Kashaev, I. Krichever, A. Losev, M. Marino, J. Morgan, H. Nakajima, D. Orlov, I. Panin, V. Pestun, J. Simons, F. Smirnov, D. Sullivan, G. Tian, M. Verbitsky.

7. Supplementary material

  • For lecture courses

Marcos Marino, An introduction to Donaldson-Witten theory

Marcos Marino, Chern-Simons Theory, Matrix Models, and Topological Strings

Sergei Gukov, Quantization via mirror symmetry

Sergei Gukov and Ingmar Saberi, Lectures on knot homology and quantum curves

N. Reshetikhin, Quasitriangular Hopf algebras and invariants of links. Algebra i Analiz, 1:2 (1989), 169188

Kashaev, R. M. R-matrix knot invariants and triangulation. Interactions between hyperbolic geometry, quantum topology and number theory, 6981, Contemp. Math., 541, Amer. Math. Sc, Providence, RI, 2011

Geer, N.; Kashaev, R.; Turaev, V.. Tetrahedral forms in monoidal categories and 3-manifold invariants. J. Reine Angew. Math. 673 (2012), 69123

Hiraku Nakajima, Geometric representation theory I

Hiraku Nakajima, Geometric representation theory II

Hiraku Nakajima, Geometric representation theory III

Jim Simons Mathematical Roots

John Morgan Topology of manifolds

John Morgan and Gang Tian, Ricci flow and Poincare conjecture

Misha Verbitsky, Hyperkahler manifolds. Lecture 1

Misha Verbitsky, Hyperkahler manifolds. Lecture 2

Misha Verbitsky, Hyperkahler manifolds. Lecture 3

  • Books on quantum mechanics and quantum field theory

L.D. Faddeev and O.A. Yakuboskii, Lectures on Quantum Mechanics for Mathematics Students

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L.A. Takhtajan, Quantum Mechanics for Mathematicians

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L.D. Faddeev and A.A. Slavnov, Gauge Fields: An Introduction To Quantum Theory

(A.A., . ., , , ., 1988)

P. Deligne and others, Eds, Quantum Fields and Strings: A Course for Mathematicians

8. Workshops and conferences:

Quantum and motivic cohomology, Fano varieties and mirror symmetry

26 - 28 September 2013

Gauge theories and integrability

3 - 5 October 2013

Conference "Frontiers and New Perspectives in Geometry and Physics"

14 - 18 October 2013